Polarization of Ideal Gas

Calculate the electric polarization P of an ideal gas, consisting of molecules having a constant electric dipole moment p in a homogeneous external electric field E at temperature τ. What is the dielectric constant of this gas at small fields?

SOLUTION

The potential energy of a dipole in an electric field E is

where the angle θ is between the direction of the electric field (which we choose to be along the ẑ axis) and the direction of a the dipole moment. The center of the spherical coordinate system is placed at the center of the dipole. The probability dw that the direction of the dipole is within a solid angle dw = sin θ dφ dθ is

(1)

The total electric dipole moment per unit volume of the gas is

(2)

Introducing a new variable x ≡ cos θ and denoting α ≡ pE/τ, we obtain

(3)

where  is the Langevin function. For pE << τ (α << 1), we can expand (3) to obtain

(4)

Since D = εE and

we have for the dielectric constant

(5)